My 7 week old baby has some mild eczema. Can we stop it from spreading further?

Gentle cleanser. You might not be able to prevent eczema from developing in other areas, but use of a gentle cleanser, like unscented dove and others, and use of a good moisturizer can gelp. 1% Hydrocortisone cream or ointment can help if the areas are red, inflasmed, or itchy.